How does the gradient increase the probabilities of the path with a positive reward in policy gradient?
Think of a surface where z-axis is $U$, x and y axis are $\theta_{1}$ and $\theta_{2}$ accordingly. Since you are following the gradient direction with respect to $\theta$ vector, that means you are moving at the direction that increases the $U$. If $R(\tau)$ is positive, you are moving towards the uphill direction and vice-versa. More formally you will say the following:   

In the policy gradient algorithm, our update step is:

$ \theta_{new} = \theta_{old} + \alpha \nabla_{\theta}U(\theta) $

So, if we select a very bad trajectory the sum of all rewards ($R(\tau)$) will be negative and the following update will shift the $\theta_{new}$ vector away from $\nabla_{\theta}U(\theta)$ vector.

If we select a good trajectory means $R(\tau)$ is positive, the update will shift $\theta_{new}$ vector towards the $\nabla_{\theta}U(\theta)$ vector. So, it will increase the probability of selecting path with positive $R$